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MA232A: Euclidean and non-EuclideanGeometry

Michaelmas Term 2015Notes on Vector Algebra and Spherical

Trigometry

David R. Wilkins

Copyright c© David R. Wilkins 2005–2015

Contents

5 Vector Algebra and Spherical Trigonometry 15.1 Vectors in Three-Dimensional Euclidean Space . . . . . . . . . 15.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 25.3 Geometrical Interpretation of the Scalar Product . . . . . . . 35.4 Geometrical Interpretation of the Vector Product . . . . . . . 55.5 Scalar Triple Products . . . . . . . . . . . . . . . . . . . . . . 85.6 The Vector Triple Product Identity . . . . . . . . . . . . . . . 105.7 Lagrange’s Quadruple Product Identity . . . . . . . . . . . . . 155.8 Orthonormal Triads of Unit Vectors . . . . . . . . . . . . . . . 165.9 Some Applications of Vector Algebra to Spherical Trigometry 17

i

5 Vector Algebra and Spherical Trigonome-

try

5.1 Vectors in Three-Dimensional Euclidean Space

A 3-dimensional vector v in the vector space R3 can be represented as a triple(v1, v2, v3) of real numbers. Vectors in R3 are added together, subtracted fromone another, and multiplied by real numbers by the usual rules, so that

(u1, u2, u3) + (v1, v2, v3) = (u1 + v1, u2 + v2, u3 + v3),

(u1, u2, u3)− (v1, v2, v3) = (u1 − v1, u2 − v2, u3 − v3),t(u1, u2, u3) = (tu1, tu2, tu3)

for all vectors (u1, u2, u3) and (v1, v2, v3) in R3, and for all real numbers t. Theoperation of vector addition is commutative and associative. Also 0+v = vfor all v ∈ R3, where 0 = (0, 0, 0), and v + (−v) = 0 for all v ∈ R3, where−(v1, v2, v3) = (−v1, v2, v3) for all (v1, v2, v3) ∈ R3. Moreover

u− v = u + (−v), t(u + v) = tu + tv, (s+ t)v = sv + tv,

s(tv) = (st)v, 1v = v

for all u,v ∈ R3 and s, t ∈ R. The set of all vectors in three-dimensionalspace, with the usual operations of vector addition and of scalar multiplica-tion constitute a three-dimensional real vector space.

The Euclidean norm |v| of a vector v is defined so that if v = (v1, v2, v3)then

|v| =√v21 + v22 + v23.

The scalar product u . v and the vector product u× v of vectors u and v aredefined such that

(u1, u2, u3) . (v1, v2, v3) = u1v1 + u2v2 + u3v3,

(u1, u2, u3)× (v1, v2, v3) = (u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1)

for all vectors (u1, u2, u3) and (v1, v2, v3) in R3. Then

(u + v) .w = u .w + v .w, u . (v + w) = u . v + u .w,

(u + v)×w = u×w + v ×w,

(tu) . v = u . (tv) = t(u . v), (tu)× v = u× (tv) = t(u× v)

u . v = v . u, u . u = |u|2, u× v = −v × u

1

for all u,v,w ∈ R3 and t ∈ R.The unit vectors i, j,k of the standard basis of R3 are defined so that

i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1).

Theni . i = j . j = k . k = 1,

i . j = j . i = j . k = k . j = k . i = i . k = 0,

i× i = j× j = k× k = 0,

i× j = −j× i = k, j× k = −k× j = i, i× j = −j× i = k.

5.2 Displacement Vectors

Let points of three-dimensional Euclidean space be represented in Cartesiancoordinates in the usual fashion, so that the line segments joining the originto the points (1, 0, 0), (0, 1, 0) and (0, 0, 1) are orthogonal and of unit lengths.

Let A and B be points in three-dimensional Euclidean space be repre-sented in Cartesian coordinates so that

A = (a1, a2, a3), B = (b1, b2, b3).

The displacement vector−→AB from A to B is defined such that

−→AB = (b1 − a1, b2 − a2, b3 − a3).

If A, B and C are points in three-dimensional Euclidean space then

−→AB +

−→BC =

−→AC.

Points A, B, C and D of three-dimensional Euclidean space are the verticesof a parallelogram (labelled in clockwise or anticlockwise) order if and only

if−→AB =

−→DC and

−→AD =

−→BC.

Let the origin O be the point with Cartesian coordinates. The positionvector of a point A (with respect to the chosen origin) is defined to be the

displacement vector−→OA.

2

5.3 Geometrical Interpretation of the Scalar Product

Let u and v be vectors in three-dimensional space, represented in some Carte-sian coordinate system by the ordered triples (u1, u2, u3) and (v1, v2, v3) re-spectively. The scalar product u . v of the vectors u and v is then given bythe formula

u . v = u1v1 + u2v2 + u3v3.

Proposition 5.1 The scalar product u . v of non-zero vectors u and v inthree-dimensional space satisfies

u . v = |u| |v| cos θ,

where θ denotes the angle between the vectors u and v.

Proof Suppose first that the angle θ between the vectors u and v is an acute

angle, so that 0 < θ < 12π. Let us consider a triangle ABC, where

−→AB = u

and−→BC = v, and thus

−→AC = u + v. Let ADC be the right-angled triangle

constructed as depicted in the figure below, so that the line AD extends ABand the angle at D is a right angle.

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��������

���

����

������

CCCCCCCCCCC

pppppppppppp pppppppppppp

pppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppp pppppppppppppppp

uu

u

u

A

B

C

D

u

vu + v

θ pppppppppppppppppppppp

Then the lengths of the line segments AB, BC, AC, BD and CD may beexpressed in terms of the lengths |u|, |v| and |u + v| of the displacementvectors u, v and u + v and the angle θ between the vectors u and v bymeans of the following equations:

AB = |u|, BC = |v|, AC = |u + v|,

3

BD = |v| cos θ and DC = |v| sin θ.

ThenAD = AB +BD = |u|+ |v| cos θ.

The triangle ADC is a right-angled triangle with hypotenuse AC. It followsfrom Pythagoras’ Theorem that

|u + v|2 = AC2 = AD2 +DC2 = (|u|+ |v| cos θ)2 + |v| sin2 θ

= |u|2 + 2|u| |v| cos θ + |v| cos2 θ + |v| sin2 θ

= |u|2 + |v|2 + 2|u| |v| cos θ,

because cos2 θ + sin2 θ = 1.Let u = (u1, u2, u3) and v = (v1, v2, v3). Then

u + v = (u1 + v1, u2 + v2, u3 + v3),

and therefore

|u + v|2 = (u1 + v1)2 + (u2 + v2)

2 + (u3 + v3)2

= u21 + 2u1v1 + v21 + u22 + 2u2v2 + v22 + u23 + 2u3v3 + v23= |u|2 + |v|2 + 2(u1v1 + u2v2 + u3v3)

= |u|2 + |v|2 + 2u . v.

On comparing the expressions for |u + v|2 given by the above equations, wesee that u . v = |u| |v| cos θ when 0 < θ < 1

2π.

The identity u . v = |u| |v| cos θ clearly holds when θ = 0 and θ = π.Pythagoras’ Theorem ensures that it also holds when the angle θ is a rightangle (so that θ = 1

2π. Suppose that 1

2π < θ < π, so that the angle θ is

obtuse. Then the angle between the vectors u and −v is acute, and is equalto π − θ. Moreover cos(π − θ) = − cos θ for all angles θ. It follows that

u . v = −u . (−v) = −|u| |v| cos(π − θ) = |u| |v| cos θ

when 12π < θ < π. We have therefore verified that the identity u . v =

|u| |v| cos θ holds for all non-zero vectors u and v, as required.

Corollary 5.2 Two non-zero vectors u and v in three-dimensional space areperpendicular if and only if u . v = 0.

Proof It follows directly from Proposition 5.1 that u . v = 0 if and only ifcos θ = 0, where θ denotes the angle between the vectors u and v. This isthe case if and only if the vectors u and v are perpendicular.

4

Example We can use the scalar product to calculate the angle θ betweenthe vectors (2, 2, 0) and (0, 3, 3) in three-dimensional space. Let u = (2, 2, 0)and v = (3, 3, 0). Then |u|2 = 22 + 22 = 8 and |v|2 = 32 + 32 = 18. It followsthat (|u| |v|)2 = 8 × 18 = 144, and thus |u| |v| = 12. Now u . v = 6. Itfollows that

6 = |u| |v| cos θ = 12 cos θ.

Therefore cos θ = 12, and thus θ = 1

3π.

We can use the scalar product to find the distance between points on asphere. Now the Cartesian coordinates of a point P on the unit sphere aboutthe origin O in three-dimensional space may be expressed in terms of anglesθ and ϕ as follows:

P = (sin θ cosϕ, sin θ sinϕ, cos θ).

The angle θ is that between the displacement vector−→OP and the vectical

vector (0, 0, 1). Thus the angle 12π−θ represents the ‘latitude’ of the point P ,

when we regard the point (0, 0, 1) as the ‘north pole’ of the sphere. Theangle ϕ measures the ‘longitude’ of the point P .

Now let P1 and P2 be points on the unit sphere, where

P1 = (sin θ1 cosϕ1, sin θ1 sinϕ1, cos θ1),

P2 = (sin θ2 cosϕ2, sin θ2 sinϕ2, cos θ2).

We wish to find the angle ψ between the displacement vectors−→OP1 and

−→OP2

of the points P1 and P2 from the origin. Now |−→OP1| = 1 and |

−→OP2| = 1. On

applying Proposition 5.1, we see that

cosψ =−→OP1 .

−→OP2

= sin θ1 sin θ2 cosϕ1 cosϕ2 + sin θ1 sin θ2 sinϕ1 sinϕ2

+ cos θ1 cos θ2

= sin θ1 sin θ2 (cosϕ1 cosϕ2 + sinϕ1 sinϕ2) + cos θ1 cos θ2

= sin θ1 sin θ2 cos(ϕ1 − ϕ2) + cos θ1 cos θ2.

5.4 Geometrical Interpretation of the Vector Product

Let a and b be vectors in three-dimensional space, with Cartesian compo-nents given by the formulae a = (a1, a2, a3) and b = (b1, b2, b3). The vectorproduct a× b is then determined by the formula

a× b = (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1).

5

Proposition 5.3 Let a and b be vectors in three-dimensional space R3.Then their vector product a × b is a vector of length |a| |b| | sin θ|, whereθ denotes the angle between the vectors a and b. Moreover the vector a× bis perpendicular to the vectors a and b.

Proof Let a = (a1, a2, a3) and b = (b1, b2, b3), and let l denote the length|a× b| of the vector a× b. Then

l2 = (a2b3 − a3b2)2 + (a3b1 − a1b3)2 + (a1b2 − a2b1)2

= a22b23 + a23b

22 − 2a2a3b2b3

+ a23b21 + a21b

23 − 2a3a1b3b1

+ a21b22 + a22b

21 − 2a1a2b1b2

= a21(b22 + b23) + a22(b

21 + b23) + a23(b

21 + b22)

− 2a2a3b2b3 − 2a3a1b3b1 − 2a1a2b1b2

= (a21 + a22 + a23)(b21 + b22 + b23)

− a21b21 − a22b22 − a23b23 − 2a2b2a3b3 − 2a3b3a1b1 − 2a1b1a2b2

= (a21 + a22 + a23)(b21 + b22 + b23)− (a1b1 + a2b2 + a3b3)

2

= |a|2|b|2 − (a . b)2

since

|a|2 = a21 + a22 + a23, |b|2 = b21 + b22 + b23, a . b = a1b1 + a2b2 + a3b3

But a . b = |a| |b| cos θ (Proposition 5.1). Therefore

l2 = |a|2|b|2(1− cos2 θ) = |a|2|b|2 sin2 θ

(since sin2 θ + cos2 θ = 1 for all angles θ) and thus l = |a| |b| | sin θ|. Also

a . (a× b) = a1(a2b3 − a3b2) + a2(a3b1 − a1b3) + a3(a1b2 − a2b1) = 0

and

b . (a× b) = b1(a2b3 − a3b2) + b2(a3b1 − a1b3) + b3(a1b2 − a2b1) = 0

and therefore the vector a × b is perpendicular to both a and b (Corol-lary 5.2), as required.

Using elementary geometry, and the formula for the length of the vectorproduct a × b given by Proposition 5.3 it is not difficult to show that thelength of this vector product is equal to the area of a parallelogram in three-dimensional space whose sides are represented, in length and direction, bythe vectors a and b.

6

Remark Let a and b be non-zero vectors that are not colinear (i.e., sothat they do not point in the same direction, or in opposite directions).The direction of a × b may be determined, using the thumb and first twofingers of your right hand, as follows. Orient your right hand such that thethumb points in the direction of the vector a and the first finger points inthe direction of the vector b, and let your second finger point outwards fromthe palm of your hand so that it is perpendicular to both the thumb andthe first finger. Then the second finger points in the direction of the vectorproduct a× b.

Indeed it is customary to describe points of three-dimensional space byCartesian coordinates (x, y, z) oriented so that if the positive x-axis andpositive y-axis are pointed in the directions of the thumb and first fingerrespectively of your right hand, then the positive z-axis is pointed in thedirection of the second finger of that hand, when the thumb and first twofingers are mutually perpendicular. For example, if the positive x-axis pointstowards the East, and the positive y-axis points towards the North, then thepositive z-axis is chosen so that it points upwards. Moreover if i = (1, 0, 0)and j = (0, 1, 0) then these vectors i and j are unit vectors pointed in thedirection of the positive x-axis and positive y-axis respectively, and i× j = k,where k = (0, 0, 1), and the vector k points in the direction of the positivez-axis. Thus the ‘right-hand’ rule for determining the direction of the vectorproduct a × b using the fingers of your right hand is valid when a = i andb = j.

If the directions of the vectors a and b are allowed to vary continuously,in such a way that these vectors never point either in the same direction or inopposite directions, then their vector product a×b will always be a non-zerovector, whose direction will vary continuously with the directions of a and b.It follows from this that if the ‘right-hand rule’ for determining the directionof a× b applies when a = i and b = j, then it will also apply whatever thedirections of a and b, since, if your right hand is moved around in such away that the thumb and first finger never point in the same direction, and ifthe second finger is always perpendicular to the thumb and first finger, thenthe direction of the second finger will vary continuously, and will thereforealways point in the direction of the vector product of two vectors pointed inthe direction of the thumb and first finger respectively.

Example We shall find the area of the parallelogram OACB in three-dimensional space, where

O = (0, 0, 0), A = (1, 2, 0), B = (−4, 2,−5), C = (−3, 4,−5).

Note that−→OC =

−→OA +

−→OB. Let a =

−→OA = (1, 2, 0) and b =

−→OB =

7

(−4, 2,−5). Then a × b = (−10, 5, 10). Now (−10, 5, 10) = 5(−2, 1, 2), and|(−2, 1, 2)| =

√9 = 3. It follows that

areaOACB = |a× b| = 15.

Note also that the vector (−2, 1, 2) is perpendicular to the parallelogramOACB.

Example We shall find the equation of the plane containing the points A, B

and C where A = (3, 4, 1), B = (4, 6, 1) and C = (3, 5, 3). Now if u =−→AB =

(1, 2, 0) and v =−→AC = (0, 1, 2) then the vectors u and v are parallel to the

plane. It follows that the vector u × v is perpendicular to this plane. Nowu × v = (4,−2, 1), and therefore the displacement vector between any twopoints of the plane must be perpendicular to the vector (4,−2, 1). It followsthat the function mapping the point (x, y, z) to the quantity 4x−2y+z mustbe constant throughout the plane. Thus the equation of the plane takes theform

4x− 2y + z = k,

for some constant k. We can calculate the value of k by substituting for x,y and z the coordinates of any chosen point of the plane. On taking thischosen point to be the point A, we find that k = 4× 3− 2× 4 + 1 = 5. Thusthe equation of the plane is the following:

4x− 2y + z = 5.

(We can check our result by verifying that the coordinates of the points A,B and C do indeed satisfy this equation.)

5.5 Scalar Triple Products

Given three vectors u, v and w in three-dimensional space, we can formthe scalar triple product u . (v ×w). This quantity can be expressed as thedeterminant of a 3× 3 matrix whose rows contain the Cartesian componentsof the vectors u, v and w. Indeed

v ×w = (v2w3 − v3w2, v3w1 − v1w3, v1w2 − v2w1),

and thus

u . (v ×w) = u1(v2w3 − v3w2) + u2(v3w1 − v1w3) + u3(v1w2 − v2w1).

8

The quantity on the right hand side of this equality defines the determinantof the 3× 3 matrix u1 u2 u3

v1 v2 v3w1 w2 w3

.

We have therefore obtained the following result.

Proposition 5.4 Let u, v and w be vectors in three-dimensional space.Then

u . (v ×w) =

∣∣∣∣∣∣u1 u2 u3v1 v2 v3w1 w2 w3

∣∣∣∣∣∣ .Corollary 5.5 Let u, v and w be vectors in three-dimensional space. Then

u . (v ×w) = v . (w × u) = w . (u× v)

= −u . (w × v) = −v . (u×w) = −w . (v × u).

Proof The basic theory of determinants ensures that 3 × 3 determinantsare unchanged under cyclic permutations of their rows by change sign undertranspositions of their rows. These identities therefore follow directly fromProposition 5.4.

One can show that the absolute value of the scalar triple product u.(v×w)is the volume of the parallelepiped in three-dimensional space whose verticesare the points whose displacement vectors from some fixed point O are 0,u, v, w, u + v, u + w, v + w and u + v + w. (A parallelepiped is a solidlike a brick, but whereas the faces of a brick are rectangles, the faces of theparallelepiped are parallelograms.)

Example We shall find the volume of the parallelepiped in 3-dimensionalspace with vertices at (0, 0, 0), (1, 2, 0), (−4, 2,−5), (0, 1, 1), (−3, 4,−5),(1, 3, 1), (−4, 3,−4) and (−3, 5,−4). The volume of this parallelepiped isthe absolute value of the scalar triple product u . (v ×w), where

u = (1, 2, 0), v = (−4, 2,−5), w = (0, 1, 1).

Now

u . (v ×w) = (1, 2, 0) . ( (−4, 2,−5)× (0, 1, 1) )

= (1, 2, 0) . (7, 4,−4) = 7 + 2× 4 = 15.

Thus the volume of the paralellepiped is 15 units.

9

5.6 The Vector Triple Product Identity

Proposition 5.6 (Vector Triple Product Identity) Let u, v and w be vectorsin three-dimensional space. Then

u× (v ×w) = (u .w)v − (u . v)w

and(u× v)×w = (u .w)v − (v .w)u.

Proof Let q = u × (v × w), and let u = (u1, u2, u3), v = (v1, v2, v3),w = (w1, w2, w3), and q = (q1, q2, q3). Then

v ×w = (v2w3 − v3w2, v3w1 − v1w3, v1w2 − v2w1).

and hence u× (v ×w) = q = (q1, q2, q3), where

q1 = u2(v1w2 − v2w1)− u3(v3w1 − v1w3)

= (u2w2 + u3w3)v1 − (u2v2 + u3v3)w1

= (u1w1 + u2w2 + u3w3)v1 − (u1v1 + u2v2 + u3v3)w1

= (u .w)v1 − (u . v)w1

Similarlyq2 = (u .w)v2 − (u . v)w2

andq3 = (u .w)v3 − (u . v)w3

(In order to verify the formula for q2 with an minimum of calculation, takethe formulae above involving q1, and cyclicly permute the subcripts 1, 2 and3, replacing 1 by 2, 2 by 3, and 3 by 1. A further cyclic permutation of thesesubscripts yields the formula for q3.) It follows that

q = (u .w)v − (u . v)w,

as required, since we have shown that the Cartesian components of the vec-tors on either side of this identity are equal. Thus

u× (v ×w) = (u .w)v − (u . v)w.

On replacing u, v and w by w, u and v respectively, we find that

w × (u× v) = (w . v)u− (w . u)v.

It follows that

(u× v)×w = −w × (u× v) = (u .w)v − (v .w)u,

as required.

10

Remark When recalling these identities for use in applications, it is oftenhelpful to check that the summands on the right hand side have the correctsign by substituting, for example, i, j and i for u, v and w, where

i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1).

Thus, for example, (i × j) × i = k × i = j and (i.i)j − (j.i)i = j. This helpscheck that the summands on the right hand side of the identity (u×v)×w =(u .w)v − (v .w)u have been chosen with the correct sign (assuming thatthese summands have opposite signs).

We present below a second proof making use of the following standardidentity.

Proposition 5.7 Let εi,j,k and δi,j be defined for i, j, k ∈ {1, 2, 3} such that

εi,j,k =

1 if (i, j, k) ∈ {(1, 2, 3), (2, 3, 1), (3, 1, 2)};−1 if (i, j, k) ∈ {(1, 3, 2), (2, 1, 3), (3, 2, 1)};

0 otherwise.

and

δi,j =

{1 if i = j;0 otherwise.

Then3∑

i=1

εi,j,k εi,m,n = δj,mδk,n − δj,nδk,m

for all i, j,m ∈ {1, 2, 3}.

Proof Suppose that j = k. Then εi,j,k = 0 for i = 1, 2, 3 and thus the lefthand side is zero. The right hand side is also zero in this case, because

δj,mδk,n − δj,nδk,m = δj,mδk,n − δk,nδj,m = 0

when j = k. Thus3∑

i=1

εi,j,k εi,m,n = δj,mδk,n − δj,nδk,m = 0 when j = k.

Similarly3∑

i=1

εi,j,k εi,m,n = δj,mδk,n − δj,nδk,m = 0 when m = n. Next suppose

that j 6= k and m 6= n but {j, k} 6= {m,n}. In this case the single valueof i in {1, 2, 3} for which εi,j,k 6= 0 does not coincide with the single value

of i for which εi,m,n 6= 0, and therefore3∑

i=1

εi,j,k εi,m,n = 0. Moreover either

j 6∈ {m,n}, in which case δj,m = δj,n = 0 and thus δj,mδk,n − δj,nδk,m = 0, or

11

else k 6∈ {m,n}, in which case δk,m = δk,n = 0 and thus δj,mδk,n−δj,nδk,m = 0.

It follows from all the cases considered above that3∑

i=1

εi,j,k εi,m,n = δj,mδk,n −

δj,nδk,m = 0 unless both j 6= k and {j, k} = {m,n}. Suppose then that j 6= kand {j, k} = {m,n}. Then there is a single value of i for which εi,j,k 6= 0.For this particular value of i we find that

εi,j,k εi,m,n =

{1 if j 6= k, j = m and k = n;−1 if j 6= k, j = n and k = m.

It follows that, in the cases where j 6= k and {j, k} = {m,n},

3∑i=1

εi,j,k εi,m,n =

1 if j 6= k, j = m and k = n,−1 if j 6= k, j = n and k = m,

0 otherwise,

= δj,mδk,n − δj,nδk,m,

as required.

Second Proof of Proposition 5.6 Let p = v × w and q = u × p =u× (v ×w), and let

u = (u1, u2, u3), v = (v1, v2, v3), w = (w1, w2, w3),

p = (p1, p2, p3) and q = (q1, q2, q3).

The definition of the vector product ensures that pi =3∑

j,k=1

εi,j,kvjwk for

i = 1, 2, 3, where εi,j,k and δi,j are defined for i, j, k ∈ {1, 2, 3} as describedin the statement of Proposition 5.7. It follows that

qm =3∑

n,i=1

εm,n,i unpi =3∑

n,i,j,k=1

εm,n,iεi,j,k unvjwk

=3∑

n,j,k=1

3∑i=1

εi,m,nεi,j,k unvjwk

=3∑

n,j,k=1

(δj,mδk,n − δj,nδk,m)unvjwk

=3∑

n,k=1

δk,n vmunwk −3∑

n,j=1

δj,n unvjwm = vm

3∑k=1

ukwk − wm

3∑j=1

ujvj

= (u .w)vm − (u . v)wm

12

for m = 1, 2, 3, and therefore

u× (v ×w) = q = (u .w)v − (u . v)w,

as required.

Remark The identity

αS . α′α′′ − α′S.α′′α = V(V . αα′ . α′′)

occurs as equation (12) in article 22 of William Rowan Hamilton’s On Quater-nions, or on a new System of Imaginaries in Algebra, published in the Philo-sophical Magazine in August 1846. Hamilton noted in that paper that thisidentity “will be found to have extensive applications.”

In Hamilton’s quaternion algebra, vectors in three-dimensional space arerepresented as pure imaginary quaternions and are denoted by Greek letters.Thus α, α′ and α′′ denote (in Hamilton’s notation) three arbitrary vectors.The product of two vectors α′ and α′′ in Hamilton’s system is a quaternionwhich is the sum of a scalar part S . αα′ and a vector part V.αα′. (The scalarand vector parts of a quaternion are the analogues, in Hamilton’s quaternionalgebra, of the real and imaginary parts of a complex number.) Now aquaternion can be represented in the form s + u1i + u2j + u3k where s, u1,u2, u3 are real numbers. The operations of quaternion addition, quaternionsubtraction and scalar multiplication by real numbers are defined so thatthe space H of quaternions is a four-dimensional vector space over the realnumbers with basis 1, i, j, k. The operation of quaternion multiplication isdefined so that quaternion multiplication is distributive over addition and isdetermined by the identities

i2 = j2 = k2 = −1,

ij = −ji = k, jk = −kj = i, ki = −ik = j

that Hamilton formulated in 1843. It then transpires that the operation ofquaternion multiplication is associative. Hamilton described his discovery ofthe quaternion algebra in a letter to P.G. Tait dated October 15, 1858 asfollows:—

. . . P.S.—To-morrow will be the 15th birthday of the Quater-nions. They started into life, or light, full grown, on [Monday] the16th of October, 1843, as I was walking with Lady Hamilton toDublin, and came up to Brougham Bridge, which my boys havesince called the Quaternion Bridge. That is to say, I then and

13

there felt the galvanic circuit of thought close; and the sparkswhich fell from it were the fundamental equations between i, j,k; exactly such as I have used them ever since. I pulled out onthe spot a pocket-book, which still exists, and made an entry, onwhich, at the very moment, I felt that it might be worth my whileto expend the labour of at least ten (or it might be fifteen) yearsto come. But then it is fair to say that this was because I felta problem to have been at that moment solved—an intellectualwant relieved—which had haunted me for at least fifteen yearsbefore.

Let quaternions q and r be defined such that q = s + u1i + u2j + u3k andr = t + v1i + v2j + v3k, where s, t, u1, u2, u3, v1, v2, v3 are real numbers. Wecan then write q = s+ α and r = t+ β, where

α = u1i+ u2j + u3k, β = v1i+ v2j + v3k.

Hamilton then defined the scalar part of the quaternion q to be the realnumber s, and the vector part of the quaternion q to be the quaternion αdetermined as described above. The Distributive Law for quaternion multi-plication and the identities for the products of i, j and k then ensure that

qr = st+ S . αβ + sβ + tα + V . αβ,

whereS . αβ = −(u1v1 + u2v2 + u3v3)

andV . αβ = (u2v3 − u3v2)i+ (u3v1 − u1v3)j + (u1v2 − u2v1)k.

Thus the scalar part S . α′α′′ of the quaternion product α′α′′ represents thenegative of the scalar product of the vectors α′ and α′′, and the vector partV.α′α′′ represents the vector product of the quaternion αα′. Thus Hamilton’sidentity can be represented, using the now customary notation for the scalarand vector products, as follows:—

−α(α′ . α′′) + α′(α′′ . α) = (α× α′)× α′′.

Hamilton’s identity of 1846 (i.e., equation (12) in article 22 of On quater-nions) is thus the Vector Triple Product Identity stated in Proposition 5.6.

Corollary 5.8 Let u, v and w be vectors in R3. Then

(u× v)× (u×w) = (u.(v ×w))u.

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Proof Using the Vector Triple Product Identity (Proposition 5.6) and basicproperties of the scalar triple product Corollary 5.5, we find that

(u× v)× (u×w) = (u.(u×w))v − (v.(u×w))u

= (u.(v ×w))u,

as required.

5.7 Lagrange’s Quadruple Product Identity

Proposition 5.9 (Lagrange’s Quadruple Product Identity) Let u, v, w andz be vectors in R3. Then

(u× v) . (w × z) = (u .w)(v . z)− (u . z)(v .w).

Proof Using the Vector Triple Product Identity (Proposition 5.6) and basicproperties of the scalar triple product Corollary 5.5, we find that

(u× v) . (w × z) = z.((u× v)×w)

= z.((u .w)v − (v .w)u)

= (u .w)(v . z)− (u . z)(v .w),

as required.

Remark Substituting i, j, i and j for u, v, w and z respectively, where

i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1),

we find that (i×j).(i×j) = k.k = 1 and (i.i)(j.j)−(i.j)(j.i) = 1−0 = 1. Thishelps check that the summands on the right hand side have been allocatedthe correct sign.

Second Proof of Proposition 5.9 Let

u = (u1, u2, u3), v = (v1, v2, v3), w = (w1, w2, w3), z = (z1, z2, z3),

and let εi,j,k and δi,j be defined for i, j, k ∈ {1, 2, 3} as described in thestatement of Proposition 5.7. Then the components of u× v are the values

of3∑

j,k=1

εi,j,kvjwk for i = 1, 2, 3. It follows from Proposition 5.7 that

(u× v) . (w × z) =∑

i,j,k,m,n

εi,j,k εi,m,n ujvkwmzn

=∑

j,k,m,n

(δj,mδk,n − δj,nδk,m)ujvkwmzn

=∑j,k

(ujvkwjzk − ujvkwkzj)

= (u .w)(v . z)− (u . z)(v .w),

15

as required.

5.8 Orthonormal Triads of Unit Vectors

Let e and f be unit vectors (i.e., vectors of length one) that are perpendicularto each other, and let g = e× f . It follows immediately from Proposition 5.3that |g| = |e| |f | = 1, and that this unit vector g is perpendicular to both eand f . Then

e . e = f . f = g . g = 1

ande . f = f . g = g . e = 0.

On applying the Vector Triple Product Identity (Proposition 5.6) we findthat

f × g = f × (e× f) = (f . f) e− (f . e) f = e,

andg × e = −e× g = −e× (e× f) = −(e . f) e + (e . e) f = f ,

Therefore

e× f = −f × e = g, f × g = −g × f = e, g × e = −e× g = f ,

Three unit vectors, such as the vectors e, f and g above, that are mutuallyperpendicular, are referred to as an orthonormal triad of vectors in three-dimensional space. The vectors e, f and g in any orthonormal triad arelinearly independent. It follows from the theory of bases and dimension infinite-dimensional vector spaces that that any vector in three-dimensionalspace may be expressed, uniquely, as a linear combination of the form

pe + qf + rg.

Any Cartesian coordinate system on three-dimensional space determinesan orthonormal triad i, j and k, where

i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1).

The scalar and vector products of these vectors satisfy the same relationsas the vectors u, v and w above. A vector represented in these Cartesiancomponents by an ordered triple (x, y, z) then satisfies the identity

(x, y, z) = xi + yj + zk.

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5.9 Some Applications of Vector Algebra to SphericalTrigometry

Let S2 be the unit sphere

{(x, y, z) ∈ R3 : x2 + y2 + z2 = 1}.

in three-dimensional Euclidean space R3. Each point of S2 may be repre-sented in the form

(sin θ cosϕ, sin θ sinϕ, cos θ).

Let I, J and K denote the points of S2 defined such that

I = (1, 0, 0), J = (0, 1, 0), K = (0, 0, 1).

We take the origin O of Cartesian coordinates to be located at the centreof the sphere. The position vectors of the points I, J and K are then thestandard unit vectors i, j and k.

It may be helpful to regard the point K as representing the “north pole”of the sphere. The “equator” is then the great circle consisting of thosepoints (x, y, z) of S2 for which z = 0. Every point P of S2 is the pole of agreat circle on S2 consisting of those points of S2 whose position vectors areorthogonal to the position vector p of the point P .

Let L and L′ be distinct points of S2 with position vectors r and r′

respectively. We denote by sinLL′ and cosLL′ the sine and cosine of theangles between the lines joining the centre of the sphere to the points L andL′.

Lemma 5.10 Let L and L′ be points on the unit sphere S2 in R3, and let rand r′ denote the displacement vectors of those points from the centre of thesphere. Then

r . r′ = cosLL′

andr× r′ = sinLL′ nL,L′ ,

where nL,L′ is a unit vector orthogonal to the plane through the centre of thesphere that contains the points L and L′.

Proof The displacement vectors r and r′ of the points L and L′ from thecentre of the sphere satisfy |r| = 1 and |r′| = 1 (because the sphere has unitradius). The required identities therefore follows from basic properties of thescalar and vector products stated in Proposition 5.1 and Proposition 5.3.

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Lemma 5.11 Let V and W be planes in R3 that are not parallel, and let nV

and vW be the unit vectors orthogonal to the planes V and W , and let α bethe angle between those planes. Then

nV . nW = cosα,

andnV × nW = sinαu,

where u is a unit vector in the direction of the line of intersection of theplanes V and W .

Proof The vectors nV and nW are not parallel, because the planes are notparallel, and therefore nV × nW is a non-zero vector. Let t = |nV × nW |.Then nV × nW = tu, where u is a unit vector orthogonal to both nV andnW . This vector u must be parallel to both V and W , and must therefore beparallel to the line of intersection of these two planes. Let v = u × nV andw = u × nW . Then the vectors v and w are parallel to the planes V andW respectively, and both vectors are orthogonal to the line of intersectionof these planes. It follows that angle between the vectors v and w is theangle α between the planes V and W .

Now the vectors v, w, nV and nW are all parallel to the plane that isorthogonal to u, the angle between the vectors v and nV is a right angle, andthe angle between the vectors w and nW is also a right angle. It follows thatthe angle between the vectors nV and nW is equal to the angle α betweenthe vectors v and w, and therefore

nV . nW = v .w = cosα,

nV × nW = v ×w = sinαu.

These identities can also be verified by vector algebra. Indeed, usingLagrange’s Quadruple Product Identity, we see that

v .w = (nV × u) . (nW × u)

= (nV . nW )(u . u)− (nV . u)(u . nW )

= nV . nW ,

because u . u = |u|2 = 1 nV . u = 0 and nW . u = 0. Thus nV . nW = cosα.Also nV × nW is parallel to the unit vector u, and therefore

v ×w = (nV × u)× (nW × u) = (u× nV )× (u× nW )

= (u.(nV × nW ))u = nV × nW .

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(see Corollary 5.8). It follows that

|nV × nW | = |v ×w| = sinα,

and thereforenV × nW = sinαu,

as required.

Proposition 5.12 (Cosine Rule of Spherical Trigonometty) Let L, L′ andL′′ be distinct points on the unit sphere in R3, let α be the angle at L betweenthe great circle through L and L′ and the great circle through L and L′′. Then

cosL′L′′ = cosLL′ · cosLL′′ + sinLL′ · sinLL′′ · cosα.

Proof The angle α at L between the great circle LL′ and the great circleLL′′ is equal to the angle between the planes through the origin that intersectthe unit sphere in those great circles, and this angle is in turn equal tothe angle between the normal vectors nL,L′ and nL,L′′ to those planes, andtherefore nL,L′ . nL,L′′ = cosα (see Lemma 5.11). Let r, r′ and r′′ denote thedisplacement vectors of the points L, L′ and L′′ respectively from the centreof the sphere. Then

r× r′ = sinLL′ nL,L′ , r× r′′ = sinLL′′ nL,L′′ .

It follows that

(r× r′).(r× r′′) = sinLL′ · sinLL′′ · cosα.

But it follows from Lagrange’s Quadruple Product Identity that Proposi-tion 5.9 that

(r× r′) . (r× r′′) = (r . r)(r′ . r′′)− (r . r′′)(r′ . r).

But r.r = |r|2 = 1, because the point r lies on the unit sphere. Therefore

(r× r′) . (r× r′′) = (r′ . r′′)− (r . r′)(r . r′′) = cosL′L′′ − cosLL′ cosLL′′.

Equating the two formulae for (r× r′) . (r× r′′), we find that

cosL′L′′ = cosLL′ · cosLL′′ + sinLL′ · sinLL′′ · cosα,

as required.

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Second Proof Let r, r′ and r′′ denote the displacement vectors of the pointsL, L′ and L′′ respectively from the centre O of the sphere. Without loss ofgenerality, we may assume that the Cartesian coordinate system with originat the centre O of the sphere has been oriented so that

r = (0, 0, 1),

r′ = (sinLL′, 0, cosLL′),

r′′ = (sinLL′′ cosα, sinLL′′ sinα, cosLL′′).

Then |r′| = 1 and |r′′| = 1. It follows that

cosL′L′′ = r′ . r′′ = cosLL′ · cosLL′′ + sinLL′ · sinLL′′ · cosα,

as required.

Theorem 5.13 (Gauss) If L, L′, L′′ and L′′′ denote four points on thesphere, and α the angle which the arcs LL′, L′′L′′′ make at their point ofintersection, then we shall have

cosLL′′ · cosL′L′′′ − cosLL′′′ · cosL′L′′ = sinLL′ · sinL′′L′′′ · cosα.

Proof Let r, r′, r′′ and r′′′ denote the displacement vectors of the pointsL, L′, L′′ and L′′′ from the centre of the sphere. It follows from Lagrange’sQuadruple Product Identity (Proposition 5.9) that

(r . r′′)(r′ . r′′′)− (r . r′′′)(r′ . r′′) = (r× r′) . (r′′ × r′′′).

Now it follows from the standard properties of the scalar and vector productsrecorded in the statement of Lemma 5.10 that r . r′′ = cosLL′′ etc., r ×r′ = sinLL′ nL,L′ and r′′ × r′′′ = sinL′′L′′′ nL′′,L′′′ , where nL,L′ is a unitvector orthogonal to the plane through the origin containing the points Land L′, and nL′′,L′′′ is a unit vector orthogonal to the plane through theorigin containing the points L′′ and L′′′. Now nL,L′ . nL′′,L′′′ = cosα, wherecosα is the cosine of the angle α between these two planes (see Lemma 5.11).This angle is also the angle, at the points of intersection, between the greatcircles on the sphere that represent the intersection of those planes with thesphere. It follows that

cosLL′′ · cosL′L′′′ − cosLL′′′ · cosL′L′′

= (r . r′′)(r′ . r′′′)− (r . r′′′)(r′ . r′′)

= (r× r′) . (r′′ × r′′′)

= sinLL′ · sinL′′L′′′ · (nL,L′ . nL′′,L′′′)

= sinLL′ · sinL′′L′′′ · cosα,

as required.

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Second Proof (This proof follows fairly closely the proof given by Gaussin the Disquisitiones Generales circa Superficies Curvas, published in 1828.)Let the point O be the centre of the sphere, and let P be the point where thegreat circle passing through LL′ intersects the great circle passing throughL′′L′′′. The angle α is then the angle between these great circles at thepoint P . Let the angles between the line OP and the lines OL, OL′, OL′′

and OL′′′ be denoted by θ, θ′, θ′′, θ′′′ respectively (so that cosPL = cos θetc.). It then follows from the Cosine Rule of Spherical Trigonometry (Propo-sition 5.12) that

cosLL′′ = cos θ cos θ′′ + sin θ sin θ′′ cosα,

cosLL′′′ = cos θ cos θ′′′ + sin θ sin θ′′′ cosα,

cosL′L′′ = cos θ′ cos θ′′ + sin θ′ sin θ′′ cosα,

cosLL′′′ = cos θ′ cos θ′′′ + sin θ′ sin θ′′′ cosα

(see Lemma 5.10). From these equations it follows that

cosLL′′ · cosL′L′′′ − cosLL′′′ · cosL′L′′

= cosα(cos θ cos θ′′ sin θ′ sin θ′′′ + cos θ′ cos θ′′′ sin θ sin θ′′

− cos θ cos θ′′′ sin θ′ sin θ′′ − cos θ′ cos θ′′ sin θ sin θ′′′)

= cosα(cos θ sin θ′ − sin θ cos θ′)(cos θ′′ sin θ′′′ − sin θ′′ cos θ′′′)

= cosα · sin(θ′ − θ) · sin(θ′′′ − θ′′)= cosα · sinLL′ · L′′L′′′,

as required.

Remark In his Disquisitiones Generales circa Superficies Curvas, publishedin 1828, Gauss proved Theorem 5.13, using the method of the second of theproofs of that theorem given above, and used it to deduce that if L, L′ andL′′ are three points on the unit sphere in R3 with Cartesian coordinates

L = (x, y, z), L′ = (x′, y′, z′), L′′ = (x′′, y′′, z′′),

and if∆ = xy′z′′ + x′y′′z + x′′yz′ − xy′′z′ − x′yz′′ − x′′y′z,

then ∆ = cosNL′′ · sinLL′, where N is a pole of the great circle passingthrough L and L′ (i.e., a point on the surface whose displacement vector fromthe sphere is orthogonal to the plane through the centre O of the sphere thatcontains the points L and L′). Now if the displacement vectors of the points

21

L, L′ and L′′ from the centre of the sphere are r, r′ and r′′ respectively thenr × r′ = sinLL′ nL,L′ , where the vector nL,L′ is orthogonal to the vectors rand r′ and has unit length. We let N be the point on the surface of the spherewhose displacement vector from the centre of the sphere is nL,L′ . Then N isa pole of the great circle passing through L and L′. It follows from this thatcosNL′′ = ± sin p, where p is the angle between the line OL′′ joining thecentre O of the sphere to L′′ and the plane through the origin that containsL and L′. It follows that

∆ = r′′ . (r× r′) = r . (cosLL′ nL,L′) = cosNL′′ · cosLL′ = ± sin p cosLL′.

Now Gauss’s paper was published nearly two decades before WilliamRowan Hamilton started publishing papers concerning vectors, using a formof vector notation that he developed in his theory of quaternions, that in-cluded standard vector identities such as those satisfied by the scalar tripleproduct, the Vector Triple Product identity and Lagrange’s Quadruple Prod-uct Identity.

Gauss deduced the identity ∆ = cosNL′′ · sinLL′ in the DisquisitionesGenerales super Superficies Curvas using the following method. Let I, J andK be the points on the surface of the sphere where the coordinate axes cutthe sphere, so that, taking the origin of Cartesian coordinates at the centreof the sphere,

I = (1, 0, 0), J = (0, 1, 0) and K = (0, 0, 1).

It then follows from an earlier theorem (Theorem 5.13 above) proved byGauss in the Disquisitiones Generales that

cosLI · cosL′J − cosLJ · cosL′I = sinLL′ · sin IJ · cosα = sinLL′ · cosα,

where α is the angle between the equatorial great circle passing through Iand J and the great circle containing L and L′ at the points of intersectionof these two circles. Now the points K and N are the poles of these twocircles, and the angle between the great circles is equal to the angle betweenthe poles of those great circles. It follows that cosα = cosNK. Also

cosLI = x, cosLJ, y, L′I = x′, L′J = y′.

It follows that xy′ − yx′ = sinLL′ · nz, where nz = cosNK. Similarlyyz′ − zy′ = sinLL′ · nx and xy′ − yz′ = sinLL′ · nz, where nx = cosNI andny = cosNJ .

∆ = (nxx′′ + nyy

′′ + nzz′′) · sinLL′ = cosNL′′ · sinLL′,

which is the identity to be proved.

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Proposition 5.14 (Gauss) Let L, L′ and L′′ be three distinct points on theunit sphere that do not all lie on any one great circle of the sphere, and letp be the angle which the line from the centre of the sphere to the point L′′

makes with the plane through the centre of the sphere that contains the pointsL and L′. Then

sin p = sinL · sinLL′′ = sinL′ · sinL′L′′,

where sinL denotes the sine of the angle between the arcs LL′ and LL′′ at Land sinL′ denotes the sine of the angle between the arcs L′L′′ and L′L at L′.

Proof Let r, r′ and r′′ denote the diplacement vectors of the points L, L′

and L′′ from the centre of the sphere. A straightforward application of theVector Triple Product Identity shows that

(r× r′)× (r× r′′) = (r.(r′ × r′′))r.

(see Corollary 5.8). Now r × r′ = sinLL′ nL,L′ , where nL,L′ is a unit vectororthogonal to the plane spanned by L and L′. Similarly r×r′′ = sinLL′′ nL,L′′ ,where nL,L′′ is a unit vector orthogonal to the plane spanned by L and L′.Moreover the vector nL,L′×nL,L′′ is orthogonal to the vectors nL,L′ and nL,L′′ ,and therefore is parallel to the line of intersection of the plane through thecentre of the sphere containing L and L′ and the plane through the centreof the sphere containing L and L′′. Moreover the magnitude of this vector isthe sine of the angle between them. It follows that nL,L′ ×nL,L′′ = ± sinL r.We note also that r.(r′× r′′) = r′′.(r× r′). (see Corollary 5.5.) Putting theseidentities together, we see that we see that

sinLL′ · sinLL′′ · sinL = ±r . (r′× r′′) = ±r′′ . (r× r′) = ± sinLL′ · r′′ . nL,L′ .

Now the cosine of the angle between the unit vector r′′ and the unit vectornL,L′ is the sine sin p of the angle between the vector r′′ and the plane throughthe centre of the sphere that contains the points L and L′. It follows thatr′′ . nL,L′ = sin p, and therefore

sinLL′ · sinLL′′ · sinL = ± sinLL′ · sin p.

Now the angles concerned are all between 0 and π, and therefore their sinesare non-negative. Also sinLL′ 6= 0, because L and L′ are distinct and arenot antipodal points on opposite sides of the sphere. Dividing by sinLL′, wefind that

sinL · sinLL′′ = sin p.

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Interchanging L and L′, we find that

sinL′ · sinL′L′′ = sin p,

as required.

Corollary 5.15 (Sine Rule of Spherical Trigonometry) Let L, L′ and L′′ bethree distinct points on the unit sphere that do not all lie on any one greatcircle of the sphere. Then

sinL′L′′

sinL=

sinLL′′

sinL′,

where sinL denotes the sine of the angle between the arcs LL′ and LL′′ at Land sinL′ denotes the sine of the angle between the arcs L′L′′ and L′L at L′.

Proposition 5.16 (Gauss) Let L, L′, L′′ be points on the unit sphere in R3,and let the point O be at the centre of that sphere. Then the volume V of thetetrahedron with apex O and base LL′L′′ satisfies

V = 16

sinL · sinLL′ · sinLL′′

= 16

sinL′ · sinLL′ · sinL′L′′

= 16

sinL′′ · sinLL′′ · sinL′L′′

where sinLL′, sinLL′′ and sinL′L′′ are the sines of the angles between thelines joining the indicated points to the centre of the sphere, and where sinL,sinL′ and sinL′′ are the sines of angles of the geodesic triangle LL′L′′ whosevertices are L and L′ and L′′ and whose sides are the arcs of great circlesjoining its vertices.

Proof This tetrahedron may be described as the tetrahedron with baseOLL′

and apex L′′. Now the area of the base of the tetrahedron is sinLL′, and theheight is sin p, where p is the perpendicular distance from the point L′′ tothe plane passing through the centre of the sphere that contains the points Land L′. The volume V of the tetrahedron is one sixth of the area of the baseof the tetrahedron multiplied by the height of the tetrahedron. On applyingProposition 5.14 we see that

V = 16

sin p · sinLL′ = 16

sinL · sinLL′ · sinLL′′.

The remaining equalities can be derived by permuting the order of the ver-tices L, L′ and L′′.

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